text: The functions sought now are periodic in the space variables of period 1. More precisely, let  be the unitary vector in the i- direction:
  Then  is periodic in the space variables if for any , then:
  Notice that this is considering the coordinates mod 1. This allows working not on the whole space  but on the quotient space , which turns out to be the 3-dimensional torus:
  Now the hypotheses can be stated properly. The initial condition  is assumed to be a smooth and divergence-free function and the external force  is assumed to be a smooth function as well. The type of solutions that are physically relevant are those who satisfy these conditions: 
  Just as in the previous case, condition 3 implies that the functions are smooth and globally defined and condition 4 means that the kinetic energy of the solution is globally bounded.